Domination Subdivision Numbers

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V − S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that 1 ≤ sdγ(G) ≤ 3 for any graph G. We give a counterexample to this conjecture. On the other hand, we show that sdγ(G) ≤ γ(G)+1 for any graph G without isolated vertices, and give constant upper bounds on sdγ(G) for several families of graphs.